Irreducible train tracks for pseudo-Anosov homeomorphisms

TL;DR

The paper proves that every pseudo-Anosov homeomorphism on a surface admits an invariant train track with an irreducible transition matrix, enabling Perron–Frobenius techniques and Markov partitions. The authors leverage veering triangulations of the mapping torus and the associated flow graph to identify obstructions to irreducibility, notably infinitesimal cycles, and then modify the train track locally to remove these obstructions. Central to the approach is constructing a layered veering triangulation, analyzing the cut-open flow graph, and proving that collapsing infinitesimal components yields a modified train track with an irreducible transition matrix. This resolves a gap in the literature and clarifies the interplay between veering geometry and train-track dynamics, with implications for robust combinatorial models of pseudo-Anosov maps.

Abstract

We describe a construction of invariant train tracks with irreducible transition matrix for pseudo-Anosov homeomorphisms. This fills what seems to be a gap in the literature concerning the existence of such train tracks. The construction starts with an invariant train track associated to the veering triangulation of the mapping torus of the homeomorphism and then uses the veering property to characterize branches which obstruct irreducibility, finally modifying the track to bypass these obstructions.

Figures (13)

• Figure 2.1: The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.
• Figure 2.2: A train track can be obtained from a finite graph embedded in $S$ by imposing a taut angle structure around each vertex, which is equivalent to having a well-defined tangent space at each vertex.
• Figure 2.3: A local picture of the duality between generic filling train tracks and triangulations of $S$. If $\tau_1$ folds to $\tau_2$, the dual triangulations differ by a flip.
• Figure 3.1: The fold from $\tau_0$ to $\tau_1$ is realized by placing a tetrahedron on $\delta_0$ above the branches being folded, so that the surface triangulation $\delta_0$ dual to $\tau_0$ is on the bottom of the resulting simplicial complex $\Delta_1$ and the surface triangulation $\delta_1$ dual to $\tau_1$ is on the top.
• Figure 3.2: The bottom edge of a tetrahedron is colored according to whether the tetrahedron is affecting a left fold or a right fold.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 2.1: A
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Proposition 3.2